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Subject: Re: Replacement of Cardinality
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Date: Wed, 21 Aug 24 12:32:30 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 21/08/2024 à 13:32, Richard Damon a écrit :
> On 8/21/24 6:44 AM, WM wrote:
>> Le 20/08/2024 à 23:25, FromTheRafters a écrit :
>>> WM explained :
>>>> Le 20/08/2024 à 12:31, FromTheRafters a écrit :
>>>>> on 8/19/2024, Richard Damon supposed :
>>>>
>>>>>> You can not derive a first number > 0 in any of the Number System 
>>>>>> that we have been talking about, Unit Fractions, Rationals or 
>>>>>> Reals, so you can't claim it to exist.
>>>>>
>>>>> Not in their natural ordering.
>>>>
>>>> Dark numbers have no discernible order. It is impossible to find the 
>>>> smallest unit fraction or the next one or the next one. It is only 
>>>> possible to prove that NUF(x) grows by 1 at every unit fraction. It 
>>>> starts from 0.
>> 
>>> Normally, the unit fractions are listed in the sequence one over one, 
>>> one over two, one over three etcetera. There is a first but no last. 
>>> Now you have started from the wrong 'end'
>> 
>> No, I have started from the other end. It exists at x > 0 because NUF(0) 
>> = 0.

> But the other end doesn't "begin" with a first Natural Number Unit 
> fraction, if it has a beginning that will be a trans-finite number.

No, it is a finite number. ∀n ∈ ℕ: 1/n - 1/(n+1) > 0 holds for all 
and only reciprocals of natural numbers.

Regards, WM